How to make object transparent on "texture" view mode
You can do it in Material viewport shading, which isn't all that different from Texture.

Set the shading to Material and add a material with the following node setup to your object.



On the Properties Editor make the following changes.



Now change the color of the RGB node to your liking.



If it's imperative that you use Texture shading, the following method should suit your needs just fine (it also works in Material shading):